Saturday, April 25, 2015

Impromptu Twitter master class on homework strategies

On Friday afternoon Anna (@borschtwithanna) tweeted out this call for conversation:

What followed was a virtual master class on handling homework in different settings. It's the sort of conversation that the #MTBoS excels at, pulling in thoughtful responses from teachers at every level of practice.

I've been thinking a lot about homework because I am overhauling my homework strategy for next year.  This blog post is my attempt to capture my strategy overview, along with pointers to resources that have helped me think through what I need to do.

Overall HW strategy for next year

There are four pillars to my homework strategy for next year:

  1.  HOMEWORK LAGS CLASSWORK: I am implementing  Henri Picciotto's strategy of having the majority of each night's homework "lag" the current classwork focus of the day; 
  2. EACH DAY'S INTRO TASK IS TABLE GROUP REVIEW: the first ten minutes of class will be for students to discuss/review/help each other out on the previous night's homework problems together in their table groups; 
  3. REFRAMING THE ROLE OF THE TEACHER: I will take questions on particularly troublesome homework problems, but I will take whole-group questions only; and 
  4. COLLECT, STAMP, & GRADE HW PACKETS EVERY 2 WEEKS: I collect, stamp, and grade homework packets every two weeks, grading for completeness of effort (every problem in every night's problem set attempted). 
Here are my elaborations on these pieces.

1. HOMEWORK LAGS CLASSWORK

Henri Picciotto's blog post on this is a classic. There are so many valuable things about having the majority of each night's homework "lag" the current day's classwork. First and foremost, it ensures that most of every night's homework is accessible to every student every day. Secondly, it helps with heterogeneous classes. It gives students multiple at-bats for each kind of homework problem, keeping things meaningful for everybody. New material challenges proficient students, while those who are still working toward proficiency get multiple opportunities to work toward master.

2. DAILY INTRO TASK IS TABLE GROUP HOMEWORK REVIEW

The first ten minutes of class are the time for students to get help on homework — and by "help" I mean helping each other first. A big function of math homework in my view is to help each student cultivate an autonomous and independent approach toward their own struggles with their own problems.

A.H. Almaas describes the problem like this: "Many people... unconsciously act out a desire to be 'saved' by a teacher. But if a teacher 'saved' you, you would lose something. You would lose the value of struggle" (Diamond Heart Book One, p. 123).  In my view, the first ten minutes of class are the place where I expect students to begin to shift their mindset about the value of struggle. If a problem is not pushing beyond their Zone of Proximal Development, I expect them to develop the habit of resolving their own confusions themselves.

3. REFRAMING THE ROLE OF THE TEACHER

In keeping with #2, I want to be conscious and intentional with framing my role in their lives about resolving their own problems with problems. Almaas describes this reframing better than anybody I know:
So there are two ways to approach the teacher. One approach is to hope the teacher will take away your problems; the other is to use the teacher, not with the expectation that she will take away your problems or offer solutions or "make it better" but that she will give you a little push in your struggle. (DH Book One, p. 124). 
This is why I love Dan Anderson's (@dandersod) description of the "mass confusion" rule: unless a problem causes mass confusion, students have to work out their problems independently and help each other out on during that first ten minutes of class.

4. COLLECT, STAMP, & GRADE HOMEWORK PACKETS EVERY TWO WEEKS

This has been the most surprisingly successful thing I've done this year. Students have to turn in a stapled packet of their "Home Enjoyment" problems every two weeks. I collect it, I stamp it, and I give them a grade for it every two weeks. Every problem must have been attempted. They should show effort to have sought a resolution for problems they didn't understand the first time through.

This has been a great accountability practice for students. It's an easy, easy 'A' grade every two weeks plus it gives them that push to keep themselves on track and not fall behind on homework. In a high-achieving school filled with motivated students, I expected not to have to do this, but in fact, my experience has revealed the reverse: students appreciate that little push toward accountability. It triggers their automatic reflexes in a way that supports their autonomy.

It also takes me very little time. I am basically just stamping packets, looking for effort and gaps, and rendering a grade whose default is 100% unless stuff is missing or late. I take off a 10% late fee per day. That is usually the only penalty students incur. I've been astounded by how being an old-school hard-ass about this has simplified and streamlined the process.

I hope this is helpful!

Sunday, April 19, 2015

The deeper wisdom of the body in math class

This post is for Malke.

As I was eavesdropping on a recent conversation between Lani Horn (@tchmathculture) and Malke Rosenfeld (@mathinyourfeet), I received a pointer to an article summarizing recent research that shows that kids with ADHD actually need to squirm in order to learn.

This makes sense to me. The deeper wisdom of the body is usually overlooked in thinking about teaching, learning, and assessment in mathematics. And yet, it can provide a vital link for our students in claiming their mathematics as well as their humanity.

I was thinking about this on Friday afternoon when I tweeted out the following:
I love the dulcet tones of compasses, rulers, & pencils during a Friday afternoon constructions quiz. #geomchat
Malke tweeted back:
I love that they are doing all that by hand. And that there are dulcet tones. :)
I responded:
Geo has affirmed my belief in the life made by hand. Huge benefits to Ss [students] from physically constructing their understanding.
I had another in a series of lightbulb moments this past month about what How People Learn says about externalizing our understanding.
And ever the good online learning partner, Malke tweeted back:
have you blogged about it?
So here I am.

In How People Learn, the authors talk about how we can use skits, presentations, and posters in group work to help students externalize their emerging understanding. This makes sense to me. In order to learn something, one first has to notice it, and that means developing a metacognitive self-awareness of the process and how it’s going.

Over the last few years, I have found that teaching students to use foldables, INBs (Interactive Notebooks), guided note-taking, and physical constructions is another extremely rich field of helping students to externalize their emerging understanding — only in these cases, they are externalizing their understanding through physical, kinesthetic processes — not just through talk, listening, and presentation processes.

The physical dimension is a good grounding for conceptual understanding. Teaching students how to literally use their tools can be a multidimensional process of making their learning both physical and tangible. Flipping open a flap or a page in a composition book is a physical manifestation of the process of retrieval or comparison or evaluation. Likewise, the process of using patty paper as a tracing medium to externalize the concept of superposition and projection of a figure to confirm congruence is a way of helping students to slow down their speeding monkey minds and to become present with the mathematics that are right in front of them.

When I tuned in to the clatter of compasses, rulers, and pencils on Friday, I really noticed how deeply engaged my students were with the geometry they were working on. Their body postures indicated how deeply immersed they were in the experience of flow: set your compass opening to an appropriate width and draw an arc across the angle you want to copy. Stab the endpoint of the segment where you want to create a new copy of your original angle and swipe the same arc there. Go back to the first angle. Refine your compass opening so that it now matches the width between the intersections of your arc and the original angle. Shift your paper, stab at the lower intersection of your copied angle-in-process and swipe an arc that will intersect with the arc you just drew there. Drop your compass; pick up your straight edge. Carefully draw a line to connect the endpoint of your target segment with the intersection of the arcs you have drawn, completing the terminal side/ ray you need to draw. Drop the straight edge; position the patty paper over your original angle and use your straight edge to trace it. Drop the straight edge and carefully slide your traced angle over your constructed angle. Does it match your figure perfectly?

The concentration etched on their brows matched the precision of their work on the page in front of them. Bisect an angle. Construct the perpendicular bisector of a segment. Construct a parallel line through an external point by using it to define an angle that you can copy.

Hopping from one stone to the next, you can cross an entire river. By placing one foot on the Earth after another in a pattern of glide reflections, you can complete a journey of a thousand miles or more.

That is one of the lessons of deductive and spatial reasoning at the heart of any good Geometry course. Noticing that it is happening in my classroom — really happening through physical, mental, and whole-hearted engagement — is one of the greatest blessings of being a teacher.


Tuesday, April 14, 2015

40 days and nights of standardized testing: a reflection

And on the 289th day, God made brownie mix. He divided the mix into the batter & the topping. And God saw the brownie mix, that it was good. And God said, Let the brownie mix be cheap and abundant and distributed to all the grocery stores so that all those who suffer, may make brownies. And the baked brownies He called "one serving," and he forbade the posting of the nutrition facts. And there were brownies, later with toppings, a comfort food.

Saturday, April 11, 2015

Resonance in our work as teachers — a love story

I went to the neighborhood farmer’s market this morning. I wanted to seek out Q and find out where she is going to college next year. I picked my way through the booths, looking for the almond grower’s stand where she works each week. After a few booths, she spotted me. She bounded over and gave me a huge hug. “I’m not selling almonds any more. I’m the assistant manager of the farner’s market now.”

I gave her another hug. “Congratulations! I’m so proud of you! What a wonderful promotion.”

“I miss you,” she said plaintively. “So does R. We talk about you all the time.”

This was a good reminder to me that we really have no idea how long or how deep our impact and reach as teachers will go. Now I know I need to find my way over to his Taco Bell / KFC Store to touch in with him.

I asked, “So where are you going to school next year?”

Her face fell. “I’m going to City College. I got rejected from all my schools.”

I hugged her again. “That’s OK. City College is awesome. You will go there for two years and transfer to the UCs or CSUs. This isn’t a defeat — it is only a setback. You are going to get where you need to go and you are going to do great. I believe in you.”

“I miss you so much,” she replied, wrapping her arms around me again.

This was all a good reminder for me. Sometimes our most important job as teachers is to show up and to be adults in their lives who are as not-full-of-shit as we can possibly be for them. We provide some much-needed continuity as adults — continuity that can be lost to them when a parent dies or moves away or goes to prison for an extended term. It is so important for all kids to have lots and lots of adults in their lives who love them in our own unique ways. We help to witness their suffering and to encourage their courage. As I write this right now, my heart hurts for her. I know how much she wanted an elite college and how disappointed she must be feeling. But I also know and believe in her unique giftedness and heroism, and I am grateful that I got a chance today to witness and reflect that back to her at a moment when she really needed it.

I will keep showing up at the farmer’s market and at KFC and believing in them both because that is an essential part of my work and my calling as a teacher. And even though Bill Gates and Arne Duncan are utterly clueless about this essential part of my work as a teacher, I will continue under all circumstances anyway. It is a great gift to get to be guerrilla bodhisattvas in our students’ lives, just as our teachers were for us.

And so I wanted to document and honor this unspoken, unappreciated, unmeasured, and undervalued part of our work as teachers on this Earth. May we all continue to be present with an open heart for all our students everywhere throughout space and time.

Friday, April 3, 2015

If it is in the way, it is the way: the only true path to a growth mindset


I believe that helping our students to find their way a growth mindset is so important it must become one of the pillars of our math teaching, but I also believe that the primary ways our leading experts are pushing right now are so misguided I can no longer stay quiet.

The bottom line is this: if you believe that a learner can simply let go of their fixed mindset just because you tell them to, then I have a bridge to sell you. I believe that the positive intentions behind this initiative are leading students to develop new ways of hiding their true selves in math class, and I can already see this approach leading to even worse forms of self-abandonment and closed-off-ness that are only going to make the whole situation much worse.

So this is my plea for us to all stop trying to coerce students into a growth mindset and instead to start developing a more mindful approach to helping students engage with a growth mindset.

Carol Dweck and Jo Boaler have done more than anyone else to popularize the idea that adopting a growth mindset is the way to go in math, but I believe that the ways they are trying to spread the gospel of a growth mindset are both harmful to students and doomed before they begin.

They are doomed because they amount to lecturing and shaming students about their defense mechanisms — an approach they would never take in the actual teaching of mathematics. A fixed mindset is a set of conditioned habits, and you can't change a habit just by force of will.

The reality is that a fixed mindset is a defense mechanism — an unconscious set of adaptive survival behaviors that evolve within a person's sense of self as a defense against what it perceives to be a threat from the outside. In the math classroom, that threat is often the threat of failure, of annihilation, of humiliation. It doesn't matter what you or I perceive the threat to be. It doesn't matter whether you or I perceive the threat to be real or not. Simply put, a fixed mindset about math — as is a self-identification as a "non-math person" —is a defense mechanism. It's not about you.

Please repeat that last part after me: a student's own personal fixed mindset about math is NOT about you.

It's the psyche's way of protecting the soft, vulnerable center of the student's own self from what it perceives to be a threat to the continued existence of its organism.

The only thing that matters in all of this is how the learner perceives the threat for him- or herself. And a fixed mindset in the learning of mathematics is a (misdirected) protective function that has arisen inside the learner as a way of keeping that learner safe from harm — often harm that you or I, as a teacher, represent.

About 40 years ago, Eugene Gendlin (the great psychotherapist from the University of Chicago) teamed up with psychologist Carl Rogers (who pioneered the humanistic or client-centered approach to psychotherapy) to investigate the question of why some people are able to make permanent and lasting change through therapy while others cannot. What Gendlin discovered was that those who make progress are the ones who are able to direct their inner "focusing" on their own subtle, internal bodily awareness or "felt sense" — a felt sense that opens the door to finding their own self-directed resolution of the problem about which they felt stuck. In his books starting with Focusing, Gendlin documented and popularized a simple yet powerful six-step process which could be taught to individuals to help them access their inner felt sense, and to work with it to bring about a "felt shift" out of their stuck place and into a freer and more authentic relationship with their triggering situations.

This process takes time and patience and psychological and emotional maturity and generosity of spirit that few of us get trained in via the usual teacher training and professional development pathways. But this is the only truly non-coercive way to support students in developing an authentic growth mindset about mathematics.

The only successful way to work with defense mechanisms — the only way that has been shown to bring about long-term inner change, either in a therapeutic or in an inner development context, such as mindfulness — involves empowering learners to gently and non-coercively notice their own defense mechanisms when they pop up.

The choice to leave behind a self-identification as a "non-math person" MUST come from inside the learner him- or herself. It cannot be imposed from the outside, no matter how well-intentioned that coercion might be.

This is what my Ignite talk at CMC-North Asilomar 2015 was about this past December. I hope this will help others to make sense of how we can best support our students on this inner development path.



Tuesday, March 24, 2015

Chords and Secants and Tangents, oh my!

Once you get to circles, chords, tangents, secants, arcs, and angles in Geometry, it's all just a nightmare of things to memorize — which means things to forget or confuse.

Rather than have students memorize this mishmash of formulas, I decided to have them focus on recognizing situations and relationships. And a good way to do that is with a four-door foldable.

Make it and use it.

Humans are tool-using animals. So let's do this thing!


Doors #1, 2, 3, and 4 each have only a diagram on their front door with color-coding because, as my brilliant colleague Alex Wilson always says, "Color tells the story."

Inside the door, we wrote as little as we possibly could. At the top we wrote our own description of the situation depicted on the door of the foldable. What's the situation? Intersecting chords. What is significant? Angle measures and arc lengths. What's the relationship? Color-code the formula.

Move on to Door #2.



We did the same thing with the chord, secant, and tangent segment theorems.




These two foldables are the only tools students can use (plus a calculator) for all of the investigations we have done with this section. They are learning to use it like a field guide — a field guide to angles and arcs, chords, secants, tangents. I hear conversation snippets like, "No, that can't be right because this is an intersecting secant and tangent situation."

My favorite is the "two tangents from an external point" set-up, which we have dubbed the "party hat situation" in which n = m (all tangents from an external point being congruent).

A big part of the success of this activity seems to come from teaching students how to make tools and to advocate for their own learning. Help them make their tools, set them loose, and get out of their way.

That's a pretty good strategy for teaching anything, I think.

Saturday, February 28, 2015

Desmos plus INBs — Conic Sections Edition

One of the things I have always been frustrated with is the crappy way example graphs look in student notebooks.

Well, no more.

For my conic sections notes sessions in Precalculus, I'm using Desmos-created graphs with all their equational and slider glory.

Here's how I ensure kids have readable, meaningful examples in their INBs:

I've created some modified graphs of the Desmos parabola graphs — one with a vertical axis and one with a horizontal axis.

I take a screen shot of the equation drawer PLUS the graph and paste it into an Omni Graffle document. For those of you playing our home game on the Mac, that's:

  • Press Cmd-Shift-4 to enter screen grab mode
  • Select the region of the Desmos window that you want to use as your graphic (this pastes it directly onto the Mac OS X clipboard)
  • Paste into a blank Omni Graffle document (from omnigroup.com )
  • Resize to fit your needs, then
  • Select, copy, and paste as many times as you need to create the master for your tiny handout
I arrange them 3-UP on the photocopy master so the tiny handout will fit onto a standards notebook/INB page.

Here are the files for my photocopy masters:
Chop, glue, annotate.

I recognize this is totally old school, but everything old is new again.